Businessman Donald Trump scored big wins Tuesday in Mississippi, Michigan and Hawaii, expanding his lead in the Republican presidential race and punching a hole in the notion that he is collapsing in the face of a barrage of attacks from fellow Republicans who are eager to kneecap his candidacy.

Networks projected Mr. Trump would win about a half-hour after polls closed at 8 p.m. in Mississippi, where nearly half of the state’s primary electorate backed him, and moments after the last polls closed in Michigan at 9 p.m. In the early morning, Mr. Trump won the Hawaii caucuses.

“There is only one person who did well tonight — Donald Trump,” Mr. Trump told supporters at his election night party in Florida.

The results, he said, showed the negative attacks against him backfired and underscored the intelligence of the electorate.

“It shows you how brilliant the public is because they knew they were lies,” said Mr. Trump, who urged the Republican establishment to unite behind him by directing their resources toward tearing down Hillary Clinton, the Democratic front-runner.

Mr. Trump took home 36.6 percent of the vote with 90 percent of the precincts reporting in Michigan. Mr. Cruz and Mr. Kasich were running neck-and-neck at about 25 percent, and Mr. Rubio had 9.3 percent.

Mr. Trump will gain the biggest chunk of the 59 delegates on the line in Michigan and 40 in Mississippi, beefing up his lead in the race, while Mr. Rubio was on track to get blanked in both those states.

The Associated Press reported there are more than 2,000 provisional votes to be counted this week, and the Hawaii Republican Party will allocate delegates after that count is completed.

Later on Tuesday night, Mr. Cruz salvaged a state and cemented his claim that he’s the only man to have beaten Mr. Trump more than once. NBC News and Fox News both projected Mr. Cruz as the winner in Idaho around midnight.

With 99 percent of the Idaho vote counted, Mr. Cruz had 45.4 percent of the vote, with Mr. Trump at 28.1 percent, Mr. Rubio of Florida at 15.9 percent and Mr. Kasich at 7.4 percent.

The four contests Tuesday coincided with an NBC News/Wall Street Journal national poll that showed Mr. Trump and Mr. Cruz locked in a tight battle for the top spot on the Republican leaderboard.

It also showed Mr. Kasich surging and Mr. Rubio slipping as the field shifts its attention toward the March 15 contests, which include winner-take-all elections in Ohio and Florida. A total of 165 delegates will be up for grabs and the fates of Mr. Kasich and Mr. Rubio will be on the line.

The contests also are emerging as the party establishment’s last best chance to stop Mr. Trump.

Victories in Ohio and Florida would keep Mr. Trump on a trajectory to win the 1,237 needed to sew up the nomination outright and avoid a contested convention, which would give his rivals another venue to try to steer the nomination away from the tycoon.

Heading into the night, Mr. Kasich hoped that a strong showing in Michigan could give him a boost heading into the Ohio primary.

“When I win Ohio, it is going to be a whole new ballgame,” Mr. Kasich said on CNN.

Tuesday’s contests came amid fresh doubts about whether some of the shine could be wearing off Mr. Trump after he scored closer-than-expected victories in Louisiana and Kentucky, and second-place finishes behind Mr. Cruz in Kansas and Maine.

Before the races were called Tuesday, Mr. Cruz told supporters in North Carolina — which also holds a March 15 primary — that he is closing the gap on Mr. Trump. He also suggested that it is time for Mr. Kasich and Mr. Rubio to leave the race, saying his rivals don’t have a “viable path to defeating Donald Trump and becoming the Republican nominee.”

Mr. Cruz urged anti-Trump voters to rally to his campaign and said the party’s chances of capturing the White House in November will be doomed with Mr. Trump atop the Republican presidential ticket.

“If we nominate Donald Trump, we lose,” he said. “Hillary beats Trump and beats him badly, which means if we nominate Donald Trump, we lose the Supreme Court for a generation, we lose the Bill of Rights for a generation, our children drown in debt.”

Mr. Trump, though, said the record-setting turnout in the nomination race shows he is broadening the GOP’s base of support.

“We have something special going on in the Republican Party and unfortunately the people in the party — they call them the ‘elites,’ or they call them whatever they call them — but those are the people that don’t respect it yet,” Mr. Trump said.

The Rubio camp, meanwhile, looked to manage expectations, saying the 44-year-old’s best states Tuesday would likely be Idaho and Hawaii, and he is focusing on the winner-take-all contest in Florida.

“We are focused on Florida,” said Rubio campaign communications director Alex Conant. “We are guaranteeing a win there. That is our priority. That is where Marco is all this week.”

Mr. Rubio also told reporters in Florida that a vote for anyone but him is the same as backing Mr. Trump. “I am the only one who can beat him in Florida,” Mr. Rubio said. “I am the only one who can stop him here.”

“These are critical times that call for a serious, thoughtful commander in chief,” Mr. Romney said. “If we Republicans were to choose Donald Trump as our nominee, the prospects for a safe and prosperous future would be greatly diminished, and I am convinced Donald Trump would lose to Hillary Clinton.”

Mr. Kasich distanced himself from the calls Tuesday afternoon, telling reporters that “Gov. Romney’s kind of recording robocalls for everyone, and I didn’t want somebody to think that he favored one person over me, because he doesn’t. You know. It’s his words. I don’t write his scripts.”

Exit polls out of Mississippi showed that Mr. Trump edged Mr. Cruz among evangelical voters. He won “angry” voters and mopped up among those looking for a political outsider.

The contests in Michigan and Mississippi attracted different electorates. Close to 8 in 10 primary voters in Mississippi identified as white evangelical and said they want a candidate who shares their beliefs. About half of voters in Michigan identified as white evangelical and just over 2 in 10 said they had a preference for someone who shared their religion.

Exit polls showed that Republican voters in Mississippi and Michigan were divided over the issue of illegal immigration, with a majority of Mississippi voters saying illegal immigrants in the U.S. should be deported, compared with most voters in Michigan who said they should be able to obtain legal status.